Theorem clnbgrssvtx 48391

Description: The closed neighborhood of a vertex 𝐾 in a graph is a subset of all vertices of the graph. (Contributed by AV, 9-May-2025.)

Hypothesis

Ref	Expression
clnbgrvtxel.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
clnbgrssvtx	⊢ (𝐺 ClNeighbVtx 𝐾) ⊆ 𝑉

Proof of Theorem clnbgrssvtx

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clnbgrvtxel.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	clnbgrisvtx 48390	. 2 ⊢ (𝑛 ∈ (𝐺 ClNeighbVtx 𝐾) → 𝑛 ∈ 𝑉)
3	2	ssriv 3931	1 ⊢ (𝐺 ClNeighbVtx 𝐾) ⊆ 𝑉

Syntax hints:   = wceq 1550   ⊆ wss 3895  ‘cfv 6506  (class class class)co 7381  Vtxcvtx 29132   ClNeighbVtx cclnbgr 48378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fv 6514  df-ov 7384  df-oprab 7385  df-mpo 7386  df-1st 7955  df-2nd 7956  df-clnbgr 48379

This theorem is referenced by:  clnbgrlevtx  48405  clnbgrisubgrgrim  48492  clnbgrgrim  48494  isubgr3stgrlem6  48531  isubgr3stgrlem7  48532  isubgr3stgrlem8  48533  isubgr3stgr  48535  uhgrimgrlim  48547  grlicref  48572  grlicsym  48573  clnbgr3stgrgrlim  48579  clnbgr3stgrgrlic  48580